Kardar-Parisi-ZhangUniversality

I. Corwin

Universality in Random SystemsUniversality in complex random systems is a strikingconcept which has played a central role in the directionof research within probability, mathematical physics andstatistical mechanics. In this article we will describe howa variety of physical systems and mathematical models,including randomly growing interfaces, certain stochas-tic PDEs, traffic models, paths in random environments,and random matrices all demonstrate the same universalstatistical behaviors in their long-time/large-scale limit.These systems are said to lie in the Kardar-Parisi-Zhang(KPZ) universality class. Proof of universality within theseclasses of systems (except for random matrices) has re-mained mostly elusive. Extensive computer simulations,nonrigorous physical arguments/heuristics, some labora-tory experiments, and limited mathematically rigorousresults provide important evidence for this belief.

Ivan Corwin is professor of mathematics at Columbia University,and research fellow of the Clay Mathematics Institute, PackardFoundation, as well as previous holder of the Poincaré Chair atthe Institut Henri Poincaré and of the Schramm Fellowship at Mi-crosoft Research and MIT.His email address is ivan.corwin@gmail.com.

For permission to reprint this article, please contact: reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1334

The last fifteen years have seen a number of break-throughs in the discovery and analysis of a handfulof special integrable probability systems which, due toenhanced algebraic structure, admit many exact com-putations and ultimately asymptotic analysis revealingthe purportedly universal properties of the KPZ class.The structures present in these systems generally origi-nate in representation theory (e.g. symmetric functions),quantum integrable systems (e.g. Bethe ansatz), alge-braic combinatorics (e.g. RSK correspondence), and thetechniques in their asymptotic analysis generally involveLaplace’s method, Fredholm determinants, or Riemann-Hilbert problem asymptotics.

This article will focus on the phenomena associatedwith the KPZ universality class [4] and highlight howcertain integrable examples expand the scopeof and refinethe notion of universality. We start by providing a briefintroduction to the Gaussian universality class and theintegrable probabilistic example of random coin flippingand the random deposition model. A small perturbationto the random deposition model leads us to the ballisticdeposition model and the KPZ universality class. Theballistic deposition model fails to be integrable; thus togain an understanding of its long-time behavior and thatof the entire KPZ class, we turn to the corner growthmodel. The rest of the article focuses on various sidesof this rich model: its role as a random growth process,

230 Notices of the AMS Volume 63, Number 3

its relation to the KPZ stochastic PDE, its interpretationin terms of interacting particle systems, and its relationto optimization problems involving paths in randomenvironments. Along the way, we include some othergeneralizationsof this processwhose integrability springsfrom the same sources. We close the article by reflectingupon some open problems.

A survey of the KPZ universality class and all ofthe associated phenomena and methods developed orutilized in its study is far too vast to be providedhere. This article presents only one of many stories andperspectives regarding this rich area of study. To evenprovide a representative cross-section of references isbeyond this scope. Additionally, though we will discussintegrable examples, we will not describe the algebraicstructures and methods of asymptotic analysis behindthem (despite their obvious importance and interest).Some recent references which review some of thesestructures include [2], [4], [8] and references therein. Onthe more physics oriented side, the collection of reviewsand books [1], [3], [5], [6], [7], [8], [9], [10] provides someidea of the scope of the study of the KPZ universalityclass and the diverse areas upon which it touches.

We start now by providing an overview of the generalnotion of universality in the context of the simplestand historically first example—fair coin flipping and theGaussian universality class.

Gaussian Universality ClassFlip a fair coin 𝑁 times. Each string of outcomes (e.g.head, tail, tail, tail, head) has equal probability 2−𝑁. Call𝐻 the (random) number of heads and let ℙ denote theprobability distribution for this sequence of coin flips.Counting shows that

ℙ(𝐻 = 𝑛) = 2−𝑁(𝑁𝑛).

Since each flip is independent, the expected number ofheads is𝑁/2. Bernoulli (1713) proved that𝐻/𝑁 convergesto 1/2 as 𝑁 goes to infinity. This was the first exampleof a law of large numbers. Of course, this does not meanthat if you flip the coin 1,000 times, you will see exactly500 heads. Indeed, in𝑁 coin flips one expects the numberof heads to vary randomly around the value 𝑁/2 in thescale √𝑁. Moreover, for all 𝑥 ∈ ℝ,

lim𝑁→∞

ℙ(𝐻 < 12𝑁+ 1

2√𝑁𝑥) =𝑥

∫−∞

𝑒−𝑦2/2

√2𝜋𝑑𝑦.

DeMoivre (1738), Gauss (1809), Adrain (1809), andLaplace(1812) all participated in the proof of this result. The lim-iting distribution is known as the Gaussian (or sometimesnormal or bell curve) distribution.

A proof of this follows from asymptotics of 𝑛!, asderived by de Moivre (1721) and named after Stirling(1729). Write

𝑛!= 𝛤(𝑛+ 1) =∞

∫0

𝑒−𝑡𝑡𝑛𝑑𝑡 = 𝑛𝑛+1∞

∫0

𝑒𝑛𝑓(𝑧)𝑑𝑧

where 𝑓(𝑧) = log𝑧 − 𝑧 and the last equality is from thechange of variables 𝑡 = 𝑛𝑧. The integral is dominated, as 𝑛grows, by the maximal value of 𝑓(𝑧) on the interval [0,∞).This occurs at𝑧 = 1, thus expanding 𝑓(𝑧)≈−1− (𝑧−1)2

2 , andplugging this into the integral yields the final expansion

𝑛!≈ 𝑛𝑛+1𝑒−𝑛√2𝜋/𝑛.This general route of writing exact formulas for prob-

abilities in terms of integrals and then performingasymptotics is quite common to the analysis of inte-grable models in the KPZ universality class, though thoseformulas and analyses are considerably more involved.

The universality of the Gaussian distribution was notbroadly demonstrated until work of Chebyshev, Markov,and Lyapunov around 1900. The central limit theorem(CLT) showed that the exact nature of coin flippingis immaterial—any sum of independent identically dis-tributed (iid) random variables with finite mean andvariance will demonstrate the same limiting behavior.

Theorem 1. Let 𝑋1,𝑋2,… be iid random variables of finitemean 𝑚 and variance 𝑣. Then for all 𝑥 ∈ ℝ,

lim𝑁→∞

ℙ(𝑁

∑𝑖=1

𝑋𝑖 < 𝑚𝑁+𝑣√𝑁𝑥) =𝑥

∫−∞

𝑒−𝑦2/2

√2𝜋𝑑𝑦.

Proofs of this result use different tools than the exactanalysis of coin flipping, and much of probability theorydeals with the study of Gaussian processes which arisethrough various generalizations of the CLT. The Gaussiandistribution is ubiquitous, and, as it is the basis for muchof classical statistics and thermodynamics, it has hadimmense societal impact.

Random versus Ballistic DepositionThe random deposition model is one of the simplest(and least realistic) models for a randomly growing one-dimensional interface. Unit blocks fall independently andin parallel from the sky above each site of ℤ accordingto exponentially distributed waiting times (see Figure 1).Recall that a random variable 𝑋 has exponential distribu-tion of rate 𝜆 > 0 (or mean 1/𝜆) if ℙ(𝑋 > 𝑥) = 𝑒−𝜆𝑥. Suchrandom variables are characterized by the memorylessproperty—conditioned on the event that 𝑋 > 𝑥,𝑋−𝑥 stillhas the exponential distribution of the same rate. Con-sequently, the random deposition model is Markov—itsfuture evolution depends only on the present state (andnot on its history).

The random deposition model is quite simple to ana-lyze, since each column grows independently. Let ℎ(𝑡, 𝑥)record the height above site 𝑥 at time 𝑡 and assumeℎ(0, 𝑥) ≡ 0. Define random waiting times 𝑤𝑥,𝑖 to be thetime for the 𝑖-th block in column 𝑥 to fall. For any 𝑛, theevent ℎ(𝑡, 𝑥) < 𝑛 is equivalent to ∑𝑛

𝑖=1 𝑤𝑥,𝑖 > 𝑡. Since the𝑤𝑥,𝑖 are iid, the law of large numbers and central limittheory apply here. Assuming 𝜆 = 1,

lim𝑡→∞

ℎ(𝑡, 𝑥)𝑡 = 1 and lim

𝑡→∞

ℎ(𝑡, 𝑥) − 𝑡𝑡1/2 ⇒ 𝑁(𝑥)

jointly over 𝑥 ∈ ℤ, where {𝑁(𝑥)}𝑥∈ℤ is a collection of iidstandard Gaussian random variables. The top of Figure 2

March 2016 Notices of the AMS 231

(a) (b)

(c) (d)

Figure 1. (A) and (B) illustrate the random depositionmodel, and (C) and (D) illustrate the ballisticdeposition model. In both cases, blocks fall fromabove each site with independent exponentiallydistributed waiting times. In the first model, theyland at the top of each column, whereas in thesecond model they stick to the first edge to whichthey become incident.

shows a simulation of the random deposition model. Thelinear growth speed and lack of spatial correlation arequite evident. The fluctuations of this model are said to bein the Gaussian universality class since they grow like 𝑡1/2,with Gaussian limit law and trivial transversal correlationlength scale 𝑡0. In general, fluctuation and transversalcorrelation exponents, as well as limiting distributions,constitute the description of a universality class, and allmodels which match these limiting behaviors are said tolie in the same universality class.

While the Gaussian behavior of this model is resilientagainst changes in the distribution of the𝑤𝑥,𝑖 (owing to theCLT), generic changes in the nature of the growth rulesshatter the Gaussian behavior. The ballistic deposition(or sticky block) model was introduced by Vold (1959)and, as one expects in real growing interfaces, displaysspatial correlation. As before, blocks fall according toiid exponential waiting times; however, now a block willstick to the first edge against which it becomes incident.This mechanism is illustrated in Figure 1. This createsoverhangs, and we define the height function ℎ(𝑡, 𝑥) asthe maximal height above 𝑥 which is occupied by a box.How does this microscopic change manifest itself overtime?

It turns out that sticky blocks radically change thelimiting behavior of this growth process. The bottom ofFigure 2 records one simulation of the process. Seppäläi-nen (1999) gave a proof that there is still an overall linear

Figure 2. Simulation of random (top) versus ballistic(bottom) deposition models driven by the sameprocess of falling blocks. The ballistic model growsmuch faster and has a smoother, more spatiallycorrelated top interface.

growth rate. Moreover, by considering a lower bound by awidth two system, one can see that this velocity exceedsthat of the random deposition model. The exact value ofthis rate, however, remains unknown.

The simulation in Figure 2 (as well as the longer timeresults displayed in Figure 3) also shows that the scale offluctuationsofℎ(𝑡, 𝑥) is smaller than in randomdepositionand that the height function remains correlated transver-sally over a long distance. There are exact conjectures forthese fluctuations. They are supposed to grow like 𝑡1/3and demonstrate a nontrivial correlation structure in atransversal scale of 𝑡2/3. Additionally, precise predictionsexist for the limiting distributions. Up to certain (presentlyundetermined) constants 𝑐1, 𝑐2, the sequence of scaledheights 𝑐2𝑡−1/3(ℎ(𝑡, 0) − 𝑐1𝑡) should converge to the so-called Gaussian orthogonal ensemble (GOE) Tracy-Widomdistributed random variable. The Tracy-Widom distribu-tions can be thought of as modern-day bell curves, andtheir names GOE or GUE (for Gaussian unitary ensemble)come from the random matrix ensembles in which thesedistributions were first observed by Tracy-Widom (1993,1994).

Ballistic deposition does not seem to be an integrableprobabilistic system, so where do these precise conjec-

232 Notices of the AMS Volume 63, Number 3

Figure 3. Simulation of random (left) versus ballistic(right) deposition models driven by the same processof falling blocks and run for a long time. The red andwhite colors represent different epochs of time in thesimulation. The size of boxes in both figures are thesame.

tures come from? The exact predictions come from theanalysis of a few similar growth processes which just hap-pen to be integrable! Ballistic deposition shares certainfeatures with these models, which are believed to be keyfor membership in the KPZ class:• Locality: Height function change depends only on

neighboring heights.• Smoothing: Large valleys are quickly filled in.• Nonlinear slope dependence: Vertical effective growth

rate depends nonlinearly on local slope.• Space-time independent noise: Growth is driven by

noise which quickly decorrelates in space and timeand does not display heavy tails.It should be made clear that a proof of the KPZ class

behavior for the ballistic deposition model is far beyondwhat can be done mathematically (though simulationsstrongly suggest that the above conjecture is true).

Corner Growth ModelWe come to the first example of an integrable probabilisticsystem in the KPZ universality class, the corner growthmodel. The randomly growing interface is modeled bya height function ℎ(𝑡, 𝑥) which is continuous, piecewiselinear, and composed of √2-length line increments ofslope +1 or −1, changing value at integer 𝑥. The heightfunction evolves according to the Markovian dynamicsthat each local minimum of ℎ (looking like ∨) turnsinto a local maximum (looking like ∧) according toan exponentially distributed waiting time. This happensindependently for each minimum. This change in heightfunction can also be thought of as adding boxes (rotated

(a) (b)

(c) (d)

Figure 4. Various possible ways that a local minimumcan grow into a local maximum. The red dotsrepresent the local minimum at which growth mayoccur.

by 45∘). See Figures 4 and 5 for further illustration of thismodel.

Wedge initial data means that ℎ(0, 𝑥) = |𝑥|, while flatinitial data (as considered for ballistic deposition) meansthat ℎ(0, 𝑥) is given by a periodic sawtooth function whichgoes between heights 0 and 1. We will focus on wedgeinitial data. Rost (1980) proved a law of large numbers forthe growing interface when time, space, and the heightfunction are scaled by the same large parameter 𝐿.Theorem 2. For wedge initial data,

lim𝐿→∞

ℎ(𝐿𝑡, 𝐿𝑥)𝐿 = 𝔥(𝑡, 𝑥) ∶= {𝑡 1−(𝑥/𝑡)2

2 |𝑥| < 𝑡,|𝑥| |𝑥| ≥ 𝑡.

Figure 6 displays the result of a computer simulationwherein the limiting parabolic shape is evident. Thefunction 𝔥 is the unique viscosity solution to the Hamilton-Jacobi equation

𝜕𝜕𝑡𝔥(𝑡, 𝑥) =

12(1 − ( 𝜕

𝜕𝑥𝔥(𝑡, 𝑥))2).

This equation actually governs the evolution of the law oflarge numbers from arbitrary initial data.

The fluctuations of this model around the law of largenumbers are believed to be universal. Figure 6 shows thatthe interface (blue) fluctuates around its limiting shape(red) on a fairly small scale, with transversal correlationon a larger scale. For 𝜖 > 0, define the scaled and centeredheight function

ℎ𝜖(𝑡, 𝑥) ∶= 𝜖𝑏ℎ(𝜖−𝑧𝑡, 𝜖−1𝑥) − 𝜖−1𝑡2

where the dynamic scaling exponent 𝑧 = 3/2 and thefluctuation exponent 𝑏 = 1/2. These exponents are easilyremembered, since they correspond with scaling time: space : fluctuations like 3 : 2 : 1. These are thecharacteristic exponents for the KPZ universality class.Johansson (1999) proved that for fixed 𝑡, as 𝜖 → 0, therandom variable ℎ𝜖(𝑡, 0) converges to a GUE Tracy-Widomdistributed random variable (see Figure 7). Results for therelated model of the longest increasing subsequence ina random permutation were provided around the sametime by Baik-Deift-Johansson (1999). For that relatedmodel, two years later Prähofer-Spohn (2001) computedthe analog to the joint distribution of ℎ𝜖(𝑡, 𝑥) for fixed 𝑡and varying 𝑥.

March 2016 Notices of the AMS 233

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 5. The corner growth model starts with anempty corner, as in (A). There is only one localminimum (the red dot), and after an exponentiallydistributed waiting time, this turns into a localmaximum by filling in the site above it with a block,as in (B). In (B) there are now two possible locationsfor growth (the two red dots). Each one has anexponentially distributed waiting time. (C)corresponds to the case when the left local minimumgrows before the right one. By the memorylessproperty of exponential random variables, once instate (C) we can think of choosing new exponentiallydistributed waiting times for the possible growthdestinations. Continuing in a similar manner, wearrive at the evolution in (D) through (H).

The entire scaled growth process ℎ𝜖(⋅, ⋅) should havea limit as 𝜖 → 0 which would necessarily be a fixedpoint under the 3:2:1 scaling. The existence of this limit(often called the KPZ fixed point) remains conjectural.Still, much is known about the properties this limitshould enjoy. It should be a stochastic process whoseevolution depends on the limit of the initial data underthe same scaling. The one-point distribution for generalinitial data, the multipoint and multitime distribution for

Figure 6. Simulation of the corner growth model. Thetop shows the model after a medium amount of time,and the bottom shows it after a longer amount oftime. The blue interface is the simulation, while thered curve is the limiting parabolic shape. The bluecurve has vertical fluctuations of order 𝑡1/3 anddecorrelates spatially on distances of order 𝑡2/3.

Figure 7. The density (top) and log of the density(bottom) of the GUE Tracy-Widom distribution.Though the density appears to look like a bell curve(or Gaussian), this comparison is misleading. Themean and variance of the distribution areapproximately −1.77 and 0.81. The tails of thedensity (as shown in terms of the log of the densityin the bottom plot) decay like 𝑒−𝑐−|𝑥|3 for 𝑥 ≪ 0 andlike 𝑒−𝑐+𝑥3/2 for 𝑥 ≫ 0 for certain positive constants 𝑐−and 𝑐+. The Gaussian density decays like 𝑒−𝑐𝑥2 in bothtails, with the constant 𝑐 related to the variance.

234 Notices of the AMS Volume 63, Number 3

wedge initial data, and various aspects of its continuity areall understood. Besides the existence of this limit, what ismissing is a useful characterization of the KPZ fixed point.Since the KPZ fixed point is believed to be the universalscaling limit of all models in the KPZ universality classand since corner growth enjoys the same key propertiesas ballistic deposition, one also is led to conjecture thatballistic deposition scales to the same fixed point andhence enjoys the same scalings and limiting distributions.The reason why the GOE Tracy-Widom distribution cameup in our earlier discussion is that we were dealing withflat rather than wedge initial data.

One test of the universality belief is to introduce partialasymmetry into the corner growth model. Now we changelocal minimum into local maximum at rate 𝑝 and turnlocal maximum into local minimum at rate 𝑞 (all waitingtimes are independent and exponentially distributed, and𝑝+𝑞 = 1). See Figure 8 for an illustration of this partiallyasymmetric corner growth model. Tracy-Widom (2007–09) showed that so long as 𝑝 > 𝑞, the same law oflarge numbers and fluctuation limit theorem hold for thepartially asymmetric model, provided that 𝑡 is replacedby 𝑡/(𝑝−𝑞). Since 𝑝−𝑞 represents the growth drift, onesimply has to speed up to compensate for this drift beingsmaller.

Clearly for 𝑝 ≤ 𝑞 something different must occurthan for 𝑝 > 𝑞. For 𝑝 = 𝑞 the law of large numbersand fluctuations change nature. The scaling of time :space : fluctuations becomes 4 : 2 : 1, and the limitingprocess under these scalings becomes the stochastic heatequation with additive white noise. This is the Edwards-Wilkinson (EW) universality class, which is described bythe stochastic heat equation with additive noise. For 𝑝 < 𝑞the process approaches a stationary distribution wherethe probability of having 𝑘 boxes added to the emptywedge is proportional to (𝑝/𝑞)𝑘.

So, we have observed that for any positive asymmetrythe growth model lies in the KPZ universality class, whilefor zero asymmetry it lies in the EW universality class. It isnatural to wonder whether in critically scaling parameters(i.e. 𝑝 − 𝑞 → 0) one might encounter a crossover regimebetween these two universality classes. Indeed, this is thecase, and the crossover is achieved by the KPZ equation,which we now discuss.

KPZ EquationThe KPZ equation is written as

𝜕ℎ𝜕𝑡 (𝑡, 𝑥) = 𝜈𝜕2ℎ

𝜕𝑥2 (𝑡, 𝑥) + 12𝜆(

𝜕ℎ𝜕𝑥 (𝑡, 𝑥))

2+√𝐷𝜉(𝑡, 𝑥),

where 𝜉(𝑡, 𝑥) is Gaussian space-time white noise; 𝜆,𝜈 ∈ ℝ;𝐷 > 0; and ℎ(𝑡, 𝑥) is a continuous function of time 𝑡 ∈ ℝ+and space 𝑥 ∈ ℝ, taking values in ℝ. Due to the whitenoise, one expects 𝑥 ↦ ℎ(𝑡, 𝑥) to be only as regular asin Brownian motion. Hence, the nonlinearity does nota priori make any sense (the derivative of Brownianmotion has negative Hölder regularity). Bertini-Cancrini(1995) provided the physically relevant notion of solution(called the Hopf-Cole solution) and showed how it arisesfrom regularizing the noise, solving the (now well-posed)

equation, and then removing the noise and subtracting adivergence.

The equation contains the four key features mentionedearlier: the growth is local, depending on the Laplacian(smoothing), the square of the gradient (nonlinear slopedependent growth), and white noise (space-time uncorre-lated noise). Kardar, Parisi, and Zhang introduced theireponymous equation and 3 : 2 : 1 scaling prediction in1986 in an attempt to understand the scaling behaviorsof random interface growth.

How might one see the 3 : 2 : 1 scaling from theKPZ equation? Define ℎ𝜖(𝑡, 𝑥) = 𝜖𝑏ℎ(𝜖−𝑧𝑡, 𝜖−1𝑥); then ℎ𝜖satisfies the KPZ equation with scaled coefficients 𝜖2−𝑧𝜈,𝜖2−𝑧−𝑏 1

2𝜆 and 𝜖𝑏− 𝑧2+

12√𝐷. It turns out that two-sided

Brownian motion is stationary for the KPZ equation;hence any nontrivial scaling must respect the Brownianscaling of the initial data and thus have 𝑏 = 1/2. Pluggingthis in, the only way to have no coefficient blow up toinfinity and not every term shrink to zero (as 𝜖 → 0) is to

It took justunder

twenty-fiveyears until Amir-Corwin-Quastel

(2010)rigorously

proved that theKPZ equation is

in the KPZuniversality

class.

choose 𝑧 = 3/2. This sug-gests theplausibility of the3 : 2 : 1 scaling. While thisheuristic gives the rightscaling, it does not providefor the scaling limit. Thelimit as 𝜖 → 0 of the equa-tion (the inviscid Burgersequation, where only thenonlinearity survives) cer-tainly does not govern thelimit of the solutions. Itremains something of amystery as to exactly howto describe this limitingKPZ fixed point. The aboveheuristic says nothing ofthe limiting distributionof the solution to the KPZequation, and there doesnot presently exist a sim-ple way to see what thisshould be.

It took just under twenty-five years until Amir-Corwin-Quastel (2010) rigorously proved that the KPZ equationis in the KPZ universality class. That work also computedan exact formula for the probability distribution of thesolution to the KPZ equation, marking the first instanceof a non-linear stochastic PDE for which this was accom-plished. Tracy-Widom’s work on the partially asymmetriccorner growthmodel and work of Bertini-Giacomin (1997)which relates that model to the KPZ equation were thetwo main inputs in this development. See [4] for fur-ther details regarding this as well as the simultaneousexact but nonrigorous steepest descent work of Sasamoto-Spohn (2010) and nonrigorous replica approach work ofCalabrese-Le Doussal-Rosso (2010) and Dotsenko (2010).

The proof that the KPZ equation is in the KPZ uni-versality class was part of an ongoing flurry of activitysurrounding the KPZ universality class from a number

March 2016 Notices of the AMS 235

(a)

(b)

Figure 8. Mapping the partially asymmetric cornergrowth model to the partially asymmetric simpleexclusion process. In (A), the red dot is a localminimum, and it grows into a maximum. In terms ofthe particle process beneath it, the minimumcorresponds to a particle followed by a hole, and thegrowth corresponds to said particle jumping into thehole to its right. In (B), the opposite is shown. Thered dot is a local maximum and shrinks into aminimum. Correspondingly, there is a hole followedby a particle, and the shrinking results in the particlemoving into the hole to its left.

of directions, such as integrable probability [4], experi-mental physics [10], and stochastic PDEs. For instance,Bertini-Cancrini’s Hopf-Cole solution relies upon a trick(the Hopf-Cole transform) which linearizes the KPZ equa-tion. Hairer (2011), who had been developing methodsto make sense of classically ill-posed stochastic PDEs, fo-cused on the KPZ equation and developed a direct notionof solution which agreed with the Hopf-Cole one but didnot require use of the Hopf-Cole transform trick. Still, thisdoes not say anything about the distribution of solutionsor their long-time scaling behaviors. Hairer’s KPZ work setthe stage for his development of regularity structures in2013—an approach to construction solutions of certaintypes of ill-posed stochastic PDEs—work for which hewas awarded a Fields Medal.

Interacting Particle SystemsThere is a direct mapping (see Figure 8) between the par-tially asymmetric corner growth model and the partiallyasymmetric simple exclusion process (generally abbrevi-ated ASEP). Associate to every −1 slope line increment aparticle on the site of ℤ above which the increment sits,and to every +1 slope line increment associate an emptysite. The height function then maps onto a configurationof particles and holes on ℤ, with at most one particle persite. When a minimum of the height function becomes a

gap = 4

rate = 1− qgap

Figure 9. The 𝑞-TASEP, whereby each particle jumpsone to the right after an exponentially distributedwaiting time with rate given by 1 − 𝑞gap.

maximum, it corresponds to a particle jumping right byone into an empty site, and likewise when a maximumbecomes a minimum, a particle jumps left by one intoan empty site. Wedge initial data for corner growth cor-responds with having all sites to the left of the origininitially occupied and all to the right empty; this is oftencalled step initial data due to the step function in terms ofparticle density. ASEP was introduced in biology literaturein 1968 by MacDonald-Gibbs-Pipkin as a model for RNA’smovement during transcription. Soon after it was inde-pendently introduced within the probability literature in1970 by Spitzer.

The earlier quoted results regarding corner growthimmediately imply that the number of particles to crossthe origin after a long time 𝑡 demonstrates KPZ classfluctuation behavior. KPZ universality would have thatgeneric changes to this model should not change the KPZclass fluctuations. Unfortunately, such generic changesdestroy the model’s integrable structure. There are afew integrable generalizations discovered in the past fiveyears which demonstrate some of the resilience of theKPZ universality class against perturbations.

TASEP (the totally asymmetric version of ASEP) is avery basic model for traffic on a one-lane road in whichcars (particles) move forward after exponential rate onewaiting times, provided the site is unoccupied. A morerealistic model would account for the fact that cars slowdown as they approach the one in front. The model of𝑞-TASEP does just that (Figure 9). Particles jump rightaccording to independent exponential waiting times ofrate 1−𝑞gap, where gap is the number of empty spaces tothe next particle to the right. Here 𝑞 ∈ [0, 1) is a differentparameter than in the ASEP, though when 𝑞 goes to zero,these dynamics become those of TASEP.

Another feature one might include in a more realistictraffic model is the cascade effect of braking. The 𝑞-pushASEP includes this (Figure 10). Particles still jumpright according to 𝑞-TASEP rules; however, now particlesmay also jump left after exponential rate 𝐿 waiting times.When such a jump occurs, it prompts the next particle tothe left to likewise jump left, with a probability given by𝑞gap, where gap is the number of empty spaces betweenthe original particle and its left neighbor. If that jumpoccurs, it may likewise prompt the next left particle tojump, and so on. Of course, braking is not the same asjumping backwards; however, if one goes into a movingframe, this left jump is like a deceleration. It turns outthat both of these models are solvable via the methodsof Macdonald processes as well as stochastic quantumintegrable systems, and thusly it has been proved that,

236 Notices of the AMS Volume 63, Number 3

rate = 1− qgaprate = L

(a)

q4

q2

(b)

Figure 10. The 𝑞-pushASEP. As shown in (A), particlesjump right according to the 𝑞-TASEP rates and leftaccording to independent exponentially distributedwaiting times of rate 𝐿. When a left jump occurs, itmay trigger a cascade of left jumps. As shown in (B),the rightmost particle has just jumped left by one.The next particle (to its left) instantaneously alsojumps left by one with probability given by 𝑞gap,where gap is the number of empty sites between thetwo particles before the left jumps occurred (in thiscase gap = 4). If that next left jump is realized, thecascade continues to the next-left particle accordingto the same rule; otherwise it stops and no otherparticles jump left in that instant of time.

just as for ASEP, they demonstrate KPZ class fluctuationbehavior (see the review [4]).

Paths in a Random EnvironmentThere is yet another class of probabilistic systems relatedto the corner growth model. Consider the totally asym-metric version of this model, started from wedge initialdata. An alternative way to track the evolving heightfunction is to record the time when a given box is grown.Using the labeling shown in Figure 11, let us call 𝐿(𝑥, 𝑦)this time, for 𝑥,𝑦 positive integers. A box (𝑥, 𝑦) may growonce its parent blocks (𝑥 − 1,𝑦) and (𝑥, 𝑦 − 1) have bothgrown, though even then it must wait for an independentexponential waiting time which we denote by 𝑤𝑥,𝑦. Thus𝐿(𝑥,𝑦) satisfies the recursion

𝐿(𝑥,𝑦) = max (𝐿(𝑥 − 1,𝑦), 𝐿(𝑥, 𝑦 − 1)) +𝑤𝑥,𝑦

subject to boundary conditions 𝐿(𝑥, 0) ≡ 0 and 𝐿(0,𝑦) ≡0. Iterating yields

𝐿(𝑥,𝑦) = max𝜋 ∑

(𝑖,𝑗)∈𝜋𝑤𝑖,𝑗

where the maximum is over all up-right and up-left latticepaths between boxes (1, 1) and (𝑥, 𝑦). This model iscalled last passage percolation with exponential weights.Following from the earlier corner growth model results,one readily sees that for any positive real (𝑥, 𝑦), for large 𝑡,𝐿(⌊𝑥𝑡⌋, ⌊𝑦𝑡⌋) demonstrates KPZ class fluctuations. A verycompelling and entirely open problem is to show that thistype of behavior persists when the distribution of the 𝑤𝑖,𝑗is no longer exponential. The only other solvable case isthat of geometric weights. A certain limit of the geometricweights leads tomaximizing the number of Poisson pointsalong directed paths. Fixing the total number of points,

w1,1

w2,1w1,2

w2,2 w3,1w1,3

w3,2 w4,1w1,4 w2,3

w4,2 w5,1

Figure 11. The relation between the corner growthmodel and last passage percolation with exponentialweights. The 𝑤𝑖,𝑗 are the waiting times between whena box can grow and when it does grow. 𝐿(𝑥,𝑦) is thetime when box (𝑥, 𝑦) grows.

this becomes equivalent to finding the longest increasingsubsequence of a random permutation. The KPZ classbehavior for this version of last passage percolation wasshown by Baik-Deift-Johansson (1999).

There is another related integrable model which canbe thought of as describing the optimal way to cross alarge grid with stoplights at intersections. Consider thefirst quadrant of ℤ2 and to every vertex (𝑥, 𝑦) assignwaiting times to the edges, leaving the vertex rightwardsand upwards. With probability 1/2 the rightward edgehas waiting time zero, while the upward edge has waitingtime given by an exponential rate 1 random variables;otherwise reverse the situation. The edge waiting timerepresents the time needed to cross an intersection inthe given direction (the walking time between lights hasbeen subtracted). The minimal passage time from (1, 1)to (𝑥, 𝑦) is given by

𝑃(𝑥,𝑦) = min𝜋 ∑

𝑒∈𝜋𝑤𝑒,

where 𝜋 goes right or up in each step and ends on thevertical line above (𝑥, 𝑦) and 𝑤𝑒 is the waiting time foredge 𝑒 ∈ 𝜋. From the origin there will always be a pathof zero waiting time whose spatial distribution is thatof the graph of a simple symmetric random walk. Justfollowing this path one can get very close to the diagonal𝑥 = 𝑦 without waiting. On the other hand, for 𝑥 ≠ 𝑦,getting to (⌊𝑥𝑡⌋, ⌊𝑦𝑡⌋) for large 𝑡 requires some amountof waiting. Barraquand-Corwin (2015) demonstrated thatas long as 𝑥 ≠ 𝑦, 𝑃(⌊𝑥𝑡⌋, ⌊𝑦𝑡⌋) demonstrates KPZ classfluctuations. This should be true when 𝜋 is restrictedto hit exactly (𝑥, 𝑦), though that result has not yet beenproved. Achieving this optimal passage time requiressome level of omnipotence, as you must be able to lookforward before choosing your route. As such, it could beconsidered as a benchmark against which to test variousrouting algorithms.

In addition to maximizing or minimizing path prob-lems, the KPZ universality class describing fluctuationsof “positive temperature” version of these models inenergetic or probabilistic favoritism is assigned to pathsbased on the sum of space-time random weights alongits graph. One such system is called directed polymers in

March 2016 Notices of the AMS 237

Figure 12. The random walk in a space time randomenvironment. For each pair of up-left and up-rightpointing edges leaving a vertex (𝑦, 𝑠), the width ofthe red edges is given by 𝑢𝑦,𝑠 and 1− 𝑢𝑦,𝑠, where 𝑢𝑦,𝑠are independent uniform random variables on theinterval [0, 1]. A walker (the yellow highlighted path)then performs a random walk in this environment,jumping up-left or up-right from a vertex withprobability equal to the width of the red edges.

random environment and is the detropicalization of LPPwhere in the definition of 𝐿(𝑥,𝑦) one replaces the opera-tions of (max,+) by (+,×). Then the resulting (random)quantity is called the partition function for the model, andits logarithm (the free energy) is conjectured for very gen-eral distributions on 𝑤𝑖,𝑗 to show KPZ class fluctuations.There is one known integrable example of weights forwhich this has been proved: the inverse-gamma distribu-tion, introduced by Seppäläinen (2009) and proved in thework by Corwin-O’Connell-Seppäläinen-Zygouras (2011)and Borodin-Corwin-Remenik (2012).

The stoplight system discussed above also has apositive temperature lifting, of which we will describe aspecial case (see Figure 12 for an illustration). For eachspace-time vertex (𝑦, 𝑠) choose a random variable 𝑢𝑦,𝑠distributed uniformly on the interval [0, 1]. Consider arandomwalk𝑋(𝑡)whichstarts at (0, 0). If the randomwalkis in position 𝑦 at time 𝑠, then it jumps to position 𝑦−1 attime 𝑠+1 with probability 𝑢𝑦,𝑠 and to position 𝑦+1 withprobability 1−𝑢𝑦,𝑠. With respect to the same environmentof 𝑢’s, consider 𝑁 such random walks. The fact that theenvironment is fixed causes them to follow certain high-probability channels. This type of system is called arandom walk in a space-time random environment, andthe behavior of a single random walker is quite wellunderstood. Let us, instead, consider the maximum of 𝑁walkers in the same environment𝑀(𝑡,𝑁) = max𝑁

𝑖=1 𝑋(𝑖)(𝑡).For a given environment, it is expected that 𝑀(𝑡,𝑁) will

localize near a given random environment dependentvalue. However, as the random environment varies, thislocalization value does as well in such a way that for 𝑟 ∈(0, 1) and large 𝑡,𝑀(𝑡, 𝑒𝑟𝑡) displays KPZ class fluctuations.

Big ProblemsIt took almost two hundred years from the discovery

KPZ universalityhas withstood

proof for almostthree decadesand shows no

signs of yielding.

of the Gaussian distribu-tions to the first proofof their universality (thecentral limit theorem). Sofar, KPZ universality haswithstoodproof for almostthree decades and showsno signs of yielding.

Besides universality,there remain a numberof other big problems forwhich little to no progresshas been made. All of the

systems and results discussed herein have been (1 + 1)-dimensional, meaning that there is one time dimensionand one space dimension. In the context of randomgrowth, it makes perfect sense (and is quite important) tostudy surface growth (1+2)-dimensional. In the isotropiccase (where the underly growth mechanism is roughlysymmetric with respect to the two spatial dimensions)there are effectively no mathematical results, though nu-merical simulations suggest that the 1/3 exponent in the𝑡1/3 scaling for corner growth should be replaced by anexponent of roughly .24. In the anisotropic case there havebeen a few integrable examples discovered which suggestvery different (logarithmic scale) fluctuations such asobserved by Borodin-Ferrari (2008).

Finally, despite the tremendous success in employingmethods of integrable probability to expand and refinethe KPZ universality class, there seems to still be quitea lot of room to grow and new integrable structuresto employ. Within the physics literature, there are anumber of exciting new directions in which the KPZ classhas been pushed, including: out-of-equilibrium transformand energy transport with multiple conservation laws,front propagation equations, quantum localization withdirected paths, and biostatistics. Equally important isunderstanding what type of perturbations break out ofthe KPZ class.

Given all of the rich mathematical predictions, onemight hope that experiments have revealed the KPZclass behavior in nature. This is quite a challenge, sincedetermining scaling exponents and limiting fluctuationsrequires immense numbers of repetition of experiments.However, there have been a few startling experimentalconfirmations of these behaviors in the context of liquidcrystal growth, bacterial colony growth, coffee stains, andfire propagation (see [10] and references therein). Truly,the study of the KPZ universality class demonstrates theunity of mathematics and physics at its best.

238 Notices of the AMS Volume 63, Number 3

Acknowledgments. The author appreciates comments ona draft of this article by A. Borodin, P. Ferrari, andH. Spohn. This text is loosely based on his “MathematicPark” lecture entitled “Universal phenomena in randomsystems”, delivered at the Institut Henri Poincaré in May2015. The author is partially supported by NSF grantDMS-1208998, by a Clay Research Fellowship, by thePoincaré chair, and by a Packard Fellowship for Scienceand Engineering.

References[1] A. Barabasi and H. Stanley, Fractal concepts in

surface growth, Cambridge University Press, 1995.[2] A. Borodin and V. Gorin, Lectures on integrable

probability, arXiv:1212.3351.[3] I. Corwin, The Kardar-Parisi-Zhang equation and uni-

versality class, Rand. Mat.: Theo. Appl. 1:1130001(2012).

[4] , Macdonald processes, quantum integrablesystems and the Kardar-Parisi-Zhang universalityclass, Proceedings of the ICM (2014).

[5] T. Halpin-Healy and Y-C. Zhang, Kinetic roughen-ing, stochastic growth, directed polymers and all that,Phys. Rep. 254:215–415 (1995).

[6] J. Krug, Origins of scale invariance in growthprocesses, Advances in Physics 46:139–282 (1997).

[7] P. Meakin, Fractals, scaling and growth far fromequilibrium, Cambridge University Press, 1998.

[8] J. Quastel and H. Spohn, The one-dimensional KPZequation and its universality class, J. Stat. Phys., toappear.

[9] H. Spohn, Large scale dynamics of interactingparticles, Springer, Heidelberg, 1991.

[10] K. Takeuchi and T. Halpin-Healy, A KPZ cocktail—shaken, not stirred…, J. Stat. Phys.Online First (2015).

Photoco

urtesy

ofCr

aigTr

acy.

Ivan Corwin

THE FEATURE COLUMNmonthly essays on mathematical topics

Each month, the Feature Column provides an online in-depth look at a mathematical topic. Complete with graphics, links, and references, the columns cover a wide spectrum of mathematics and its applications, often including historical figures and their contributions. The authors—David Austin, Bill Casselman, Joe Malkevitch, and Tony Phillips—share their excitement about developments in math-ematics.

Recent essays include:

Mathematics and Ecology

Game. SET. Line.

Sums and Integrals: The Swiss analysis knife

Math and the Sewing Machine

Mathematical Careers

The Stable Marriage Problemand School Choice

The Mathematics of the Rainbow, Part II

Mathematics and Psychology

www.ams.org/featurecolumn

AMERICAN MATHEMATICAL SOCIETY

Subscribe to an RSS feed for each new essay at ams.org/featurecolumn.rss

March 2016 Notices of the AMS 239